1,063 research outputs found
A mathematical formalization of the parallel replica dynamics
The purpose of this article is to lay the mathematical foundations of a well
known numerical approach in computational statistical physics and molecular
dynamics, namely the parallel replica dynamics introduced by A.F. Voter. The
aim of the approach is to efficiently generate a coarse-grained evolution (in
terms of state-to-state dynamics) of a given stochastic process. The approach
formally consists in concurrently considering several realizations of the
stochastic process, and tracking among the realizations that which, the
soonest, undergoes an important transition. Using specific properties of the
dynamics generated, a computational speed-up is obtained. In the best cases,
this speed-up approaches the number of realizations considered. By drawing
connections with the theory of Markov processes and, in particular, exploiting
the notion of quasi-stationary distribution, we provide a mathematical setting
appropriate for assessing theoretically the performance of the approach, and
possibly improving it
The parallel replica method for simulating long trajectories of Markov chains
The parallel replica dynamics, originally developed by A.F. Voter,
efficiently simulates very long trajectories of metastable Langevin dynamics.
We present an analogous algorithm for discrete time Markov processes. Such
Markov processes naturally arise, for example, from the time discretization of
a continuous time stochastic dynamics. Appealing to properties of
quasistationary distributions, we show that our algorithm reproduces exactly
(in some limiting regime) the law of the original trajectory, coarsened over
the metastable states.Comment: 13 pages, 6 figure
Two mathematical tools to analyze metastable stochastic processes
We present how entropy estimates and logarithmic Sobolev inequalities on the
one hand, and the notion of quasi-stationary distribution on the other hand,
are useful tools to analyze metastable overdamped Langevin dynamics, in
particular to quantify the degree of metastability. We discuss the interest of
these approaches to estimate the efficiency of some classical algorithms used
to speed up the sampling, and to evaluate the error introduced by some
coarse-graining procedures. This paper is a summary of a plenary talk given by
the author at the ENUMATH 2011 conference
A walk in the statistical mechanical formulation of neural networks
Neural networks are nowadays both powerful operational tools (e.g., for
pattern recognition, data mining, error correction codes) and complex
theoretical models on the focus of scientific investigation. As for the
research branch, neural networks are handled and studied by psychologists,
neurobiologists, engineers, mathematicians and theoretical physicists. In
particular, in theoretical physics, the key instrument for the quantitative
analysis of neural networks is statistical mechanics. From this perspective,
here, we first review attractor networks: starting from ferromagnets and
spin-glass models, we discuss the underlying philosophy and we recover the
strand paved by Hopfield, Amit-Gutfreund-Sompolinky. One step forward, we
highlight the structural equivalence between Hopfield networks (modeling
retrieval) and Boltzmann machines (modeling learning), hence realizing a deep
bridge linking two inseparable aspects of biological and robotic spontaneous
cognition. As a sideline, in this walk we derive two alternative (with respect
to the original Hebb proposal) ways to recover the Hebbian paradigm, stemming
from ferromagnets and from spin-glasses, respectively. Further, as these notes
are thought of for an Engineering audience, we highlight also the mappings
between ferromagnets and operational amplifiers and between antiferromagnets
and flip-flops (as neural networks -built by op-amp and flip-flops- are
particular spin-glasses and the latter are indeed combinations of ferromagnets
and antiferromagnets), hoping that such a bridge plays as a concrete
prescription to capture the beauty of robotics from the statistical mechanical
perspective.Comment: Contribute to the proceeding of the conference: NCTA 2014. Contains
12 pages,7 figure
The Relativistic Hopfield network: rigorous results
The relativistic Hopfield model constitutes a generalization of the standard
Hopfield model that is derived by the formal analogy between the
statistical-mechanic framework embedding neural networks and the Lagrangian
mechanics describing a fictitious single-particle motion in the space of the
tuneable parameters of the network itself. In this analogy the cost-function of
the Hopfield model plays as the standard kinetic-energy term and its related
Mattis overlap (naturally bounded by one) plays as the velocity. The
Hamiltonian of the relativisitc model, once Taylor-expanded, results in a
P-spin series with alternate signs: the attractive contributions enhance the
information-storage capabilities of the network, while the repulsive
contributions allow for an easier unlearning of spurious states, conferring
overall more robustness to the system as a whole. Here we do not deepen the
information processing skills of this generalized Hopfield network, rather we
focus on its statistical mechanical foundation. In particular, relying on
Guerra's interpolation techniques, we prove the existence of the infinite
volume limit for the model free-energy and we give its explicit expression in
terms of the Mattis overlaps. By extremizing the free energy over the latter we
get the generalized self-consistent equations for these overlaps, as well as a
picture of criticality that is further corroborated by a fluctuation analysis.
These findings are in full agreement with the available previous results.Comment: 11 pages, 1 figur
Free energies of Boltzmann Machines: self-averaging, annealed and replica symmetric approximations in the thermodynamic limit
Restricted Boltzmann machines (RBMs) constitute one of the main models for
machine statistical inference and they are widely employed in Artificial
Intelligence as powerful tools for (deep) learning. However, in contrast with
countless remarkable practical successes, their mathematical formalization has
been largely elusive: from a statistical-mechanics perspective these systems
display the same (random) Gibbs measure of bi-partite spin-glasses, whose
rigorous treatment is notoriously difficult. In this work, beyond providing a
brief review on RBMs from both the learning and the retrieval perspectives, we
aim to contribute to their analytical investigation, by considering two
distinct realizations of their weights (i.e., Boolean and Gaussian) and
studying the properties of their related free energies. More precisely,
focusing on a RBM characterized by digital couplings, we first extend the
Pastur-Shcherbina-Tirozzi method (originally developed for the Hopfield model)
to prove the self-averaging property for the free energy, over its quenched
expectation, in the infinite volume limit, then we explicitly calculate its
simplest approximation, namely its annealed bound. Next, focusing on a RBM
characterized by analogical weights, we extend Guerra's interpolating scheme to
obtain a control of the quenched free-energy under the assumption of replica
symmetry: we get self-consistencies for the order parameters (in full agreement
with the existing Literature) as well as the critical line for ergodicity
breaking that turns out to be the same obtained in AGS theory. As we discuss,
this analogy stems from the slow-noise universality. Finally, glancing beyond
replica symmetry, we analyze the fluctuations of the overlaps for an estimate
of the (slow) noise affecting the retrieval of the signal, and by a stability
analysis we recover the Aizenman-Contucci identities typical of glassy systems.Comment: 21 pages, 1 figur
Anergy in self-directed B lymphocytes from a statistical mechanics perspective
The ability of the adaptive immune system to discriminate between self and
non-self mainly stems from the ontogenic clonal-deletion of lymphocytes
expressing strong binding affinity with self-peptides. However, some
self-directed lymphocytes may evade selection and still be harmless due to a
mechanism called clonal anergy. As for B lymphocytes, two major explanations
for anergy developed over three decades: according to "Varela theory", it stems
from a proper orchestration of the whole B-repertoire, in such a way that
self-reactive clones, due to intensive interactions and feed-back from other
clones, display more inertia to mount a response. On the other hand, according
to the `two-signal model", which has prevailed nowadays, self-reacting cells
are not stimulated by helper lymphocytes and the absence of such signaling
yields anergy. The first result we present, achieved through disordered
statistical mechanics, shows that helper cells do not prompt the activation and
proliferation of a certain sub-group of B cells, which turn out to be just
those broadly interacting, hence it merges the two approaches as a whole (in
particular, Varela theory is then contained into the two-signal model). As a
second result, we outline a minimal topological architecture for the B-world,
where highly connected clones are self-directed as a natural consequence of an
ontogenetic learning; this provides a mathematical framework to Varela
perspective. As a consequence of these two achievements, clonal deletion and
clonal anergy can be seen as two inter-playing aspects of the same phenomenon
too
- …